19. Show that the function defined by g (x) = x- [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

Answers (1)

Given function is
g (x) = x- [x]
Given is defined for all real numbers k
\lim_{x\rightarrow k^-}f(x) = k - (k-1) = k-k+1 =1\\ \lim_{x\rightarrow k^+}f(x) = k - k = 0\\ \lim_{x\rightarrow k^-}f(x) \neq \lim_{x\rightarrow k^+}f(x)
Hence, by this, we can say that  the function defined by g (x) = x- [x] is discontinuous at all integral points

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